Partitions
All data on drive X may be permanently destroyed by using the following command (the drive must be unmounted first, which means that a LiveDVD or LiveUSB is useful in this case): warning - this will take a very long time

dd if=/dev/zero of=/dev/sdX

For the truly paranoid with plenty of time to spare:

dd if=/dev/random of=/dev/sdX && dd if=/dev/zero of=/dev/sdX

Single files
Single files may be made permanently inaccessible by encrypting them before deleting them in the usual way.